\begin{answer}
    In this case
$$
l(W) = \sum_{i=1}^m(\log |W| + \sum_{j=1}^m(-\log 2 - |w^T_jx^{(i)}|))
$$
and
$$
\begin{aligned}
\nabla_Wl(W) &= \sum_{i=1}^m (W^{-T} -  l^{(i)}x^{(i)T})
\end{aligned}
$$
Where $l^{(i)}$ is an indicator vector:
$$
l^{(i)}_j = \begin{cases}
1, &w_j^Tx^{(i)} >0\\
-1, &w^T_jx^{(i)}\le 0
\end{cases}
$$
And we can simplify this to
$$
\nabla_Wl(W) = mW^{-T} - LX
$$
where the $i$-th column of $L$ is $l^{(i)}$. And the update rule for a single example:
$$
W:= W + \alpha(W^{-T}  - lx^T)
$$
where $l$ is as defined above.
\end{answer}
